Fourier Coefficient

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Fourier Coefficient


Fourier coefficients are the coefficients

in the Fourier series expansion of a periodic function f(x) with period 2Ƭ (see). Formulas (*) are sometimes called the Euler-Fourier formulas.

A continuous function f(x) is uniquely determined by its Fourier coefficients. The Fourier coefficients of an integrable function f(x) approach zero as n → ∞. Moreover, the rate of their decrease depends on the differentiability properties of f(x). For example, if f(x) has k continuous derivatives, then there is a number c such that |an| ≤ clnk and |bn| ≤ clnk. The Fourier coefficients are also connected with f(x) by the equality

(seePARSEVAL EQUALITY). The Fourier coefficients of a function f(x) with respect to any normalized system of functions ϕ1(x), ϕ2, . . ., ϕn(x), . . . orthogonal on a segment [a, b] are given by the formula


The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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